In the late seventies, Clark [In Communication Systems and Random Process Theory (Proc. 2nd NATO Advanced Study Inst., Darlington, 1977) (1978) 721-734, Sijthoff & Noordhoff] pointed out that it would be natural for $\pi_t$, the solution of the stochastic filtering problem, to depend continuously on the observed data $Y=\{Y_s,s\in[0,t]\}$. Indeed, if the signal and the observation noise are independent one can show that, for any suitably chosen test function $f$, there exists a continuous map $\theta^f_t$, defined on the space of continuous paths $C([0,t],\mathbb{R}^d)$ endowed with the uniform convergence topology such that $\pi_t(f)=\theta^f_t(Y)$, almost surely; see, for example, Clark [In Communication Systems and Random Process Theory (Proc. 2nd NATO Advanced Study Inst., Darlington, 1977) (1978) 721-734, Sijthoff & Noordhoff], Clark and Crisan [Probab. Theory Related Fields 133 (2005) 43-56], Davis [Z. Wahrsch. Verw. Gebiete 54 (1980) 125-139], Davis [Teor. Veroyatn. Primen. 27 (1982) 160-167], Kushner [Stochastics 3 (1979) 75-83]. As shown by Davis and Spathopoulos [SIAM J. Control Optim. 25 (1987) 260-278], Davis [In Stochastic Systems: The Mathematics of Filtering and Identification and Applications, Proc. NATO Adv. Study Inst. Les Arcs, Savoie, France 1980 505-528], [In The Oxford Handbook of Nonlinear Filtering (2011) 403-424 Oxford Univ. Press], this type of robust representation is also possible when the signal and the observation noise are correlated, provided the observation process is scalar. For a general correlated noise and multidimensional observations such a representation does not exist. By using the theory of rough paths we provide a solution to this deficiency: the observation process $Y$ is "lifted" to the process $\mathbf{Y}$ that consists of $Y$ and its corresponding L\'{e}vy area process, and we show that there exists a continuous map $\theta_t^f$, defined on a suitably chosen space of H\"{o}lder continuous paths such that $\pi_t(f)=\theta_t^f(\mathbf{Y})$, almost surely.Comment: Published in at http://dx.doi.org/10.1214/12-AAP896 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Backward stochastic differential equations (BSDEs) in the sense of Pardoux-Peng [Backward stochastic differential equations and quasilinear parabolic partial differential equations, Lecture Notes in Control and Inform. Sci., 176, 200-217, 1992] provide a non-Markovian extension to certain classes of nonlinear partial differential equations; the non-linearity is expressed in the so-called driver of the BSDE. Our aim is to deal with drivers which have very little regularity in time. To this end we establish continuity of BSDE solutions with respect to rough path metrics in the sense of Lyons [Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14, no. 2, 215-310, 1998] and so obtain a notion of "BSDE with rough driver". Existence, uniqueness and a version of Lyons' limit theorem in this context are established. Our main tool, aside from rough path analysis, is the stability theory for quadratic BSDEs due to Kobylanski [Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab., 28(2):558-602, 2000].

A. We give meaning and study the regularity of di erential equations with a rough path term and a Brownian noise term, that is we are interested in equations of the typewhere η is a deterministic geometric, step-2 rough path and B is a multi-dimensional Brownian motion. En passant, we give a short and direct argument that implies integrability estimates for rough di erential equations with Gaussian driving signals which is of independent interest. IThe contribution of this article is twofold: rstly, we give meaning to di erential equations of the typethat is, for a deterministic, step 2-rough path η we are looking for a stochastic process S η that is adapted to σ (B) and study the regularity of the map η → S η . Secondly, we take this as an opportunity to revisit the integrability estimates of solutions of rough di erential equations driven by Gaussian processes. If either b ≡ 0 or c ≡ 0 then rough path theory [26,27,29,19,18] or standard Itōcalculus allow (under appropriate regularity assumptions on the vector elds (a, b, c)) to give meaning to (1.1). However, in the generic case when the vector elds b and c have a non-trivial Lie bracket, any notion of a solution (that is consistent with an Itō-Stratonovich calculus) must take into account the area swept out between the trajectories of B and η. A natural approach is to identify S η as the RDE solution of2000 Mathematics Subject Classi cation. 60H10,60H30,60H05,60G15,60G17. Key words and phrases. Existence of path integrals, Integrability of rough di erential equations with Gaussian signals, Clark's robustness problem in nonlinear ltering, Viscosity solutions of RPDEs. 1 A LEVY-AREA BETWEEN BROWNIAN MOTION AND ROUGH PATHS WITH APPLICATIONS 2where Λ is a joint, step-2 rough path lift between the enhanced Brownian motion B = 1 + B +´B ⊗ •dB and η, and (r, Λ) is the joint rough path between the random rough path Λ and the bounded variation path r → r. While the existence of a joint lift between a continuous bounded variation path and any rough path is trivial (via integration by parts), the existence of a joint lift between two given step-2 rough paths is more subtle and in general not possible. More precisely, let α ∈ 1 3 , 1 2 and denote with C 0,α R d the space of geometric, step-2, α-Hölder rough paths over R d (we often only write C 0,α and d is chosen according to context). Fix two geometric, step-2 rough paths η = (1 + ηIn general, one cannot hope to nd a joint rough path lift, i.e. a geometric rough path λ = 1 + λ 1 + λ 2 ∈ C 0,α R d+e such that (formally)since the entries on the cross-diagonal of λ 2 are not well-de ned. (What is guaranteed by the extension theorem in [28] is that there exists a weak geometric rough path λ such that λ 1 = η 1 , b 1 , however this λ is highly non-unique and no consistency with η or b on the second level is guaranteed).In Section 2 we show that in the case when the deterministic rough path b is replaced by enhanced Brownian motion B, there does indeed exists a stochastic process Λ which merits in a certain sense to be called...

We consider a system of infinitely many interacting Brownian motions that models the height of a one-dimensional interface between two bulk phases. We prove that the large scale fluctuations of the system are well approximated by the solution to the KPZ equation provided the microscopic interaction is weakly asymmetric. The proof is based on the martingale solutions of Gonçalves and Jara [GJ14] and the corresponding uniqueness result of [GP15a].